Abstract. In this paper geometric phases (Berry and Aharonov-Bohm) are generalized
to nonlinear topological phase fields on pseudospheres, where the
coordinate vector field is parallel transported along the
signal/soliton vector field with Levi--Civita connection.
Projective $PSL(2,{\Bbb R})$ symmetry describes the relativistic
self-interacting bosonic sine-Gordon field. A Coulomb potential
can be induced as the stereographic projection of a harmonic
oscillator potential mapping angles or phases to distances and
vice versa resulting in mutual coupling with a generalized
coupling constant given by a nonlinear iteration. With
single-valuedness requirement in 137-gonal symmetry it fits within
a few ppb uncertainty to the Sommerfeld fine structure constant.